3-2-x-3-2-x-10-Solve-for-x-

Question Number 215992 by MATHEMATICSAM last updated on 24/Jan/25

{(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10 . Solve

for x .

Answered by Rasheed.Sindhi last updated on 24/Jan/25

{(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10 . Solve for x .

\sqrt{3} + \sqrt{2} = a

\frac{1}{a} = \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \sqrt{3} - \sqrt{2}

a^{x} + \frac{1}{a^{x}} = 10

a^{2 x} - 10 a^{x} + 1 = 0

a^{x} = \frac{10 \pm \sqrt{100 - 4}}{2} = 5 \pm 2 \sqrt{6}

x \log a = \log (5 \pm 2 \sqrt{6})

x = \frac{\log (5 \pm 2 \sqrt{6})}{\log (a)}

= \frac{\log (5 \pm 2 \sqrt{6})}{\log (\sqrt{3} + \sqrt{2})}

x = \log_{(\sqrt{3} + \sqrt{2})} (5 \pm 2 \sqrt{6})

= \log_{(\sqrt{3} + \sqrt{2})} {(\sqrt{3} \pm \sqrt{2})}^{2}

= 2, - 2